UNIPLANAR MOTION OF A RIGID BODY 235 Therefore after the time k the hoop will roll along towards the right with a linear velocity v2 less than VQ, and with an angular velocity o> greater than OJQ. Case III. — Suppose the initial rotation to be counter-clockwise. In this case we obtain V3 _ s^a, (9,0 where U is the time when sliding ceases. There are three special cases to be considered : (a) When VQ > aa>0, vz is positive, and consequently the hoop goes on rolling towards the right. (b) When VQ = aco0, vs = 0, and consequently at t = tz the hoop comes to rest. (c) When VQ < oco0, v$ is negative. Therefore at the instant t = t& the hoop begins to roll backwards. PROBLEMS. 1. Discuss the motion of the following bodies rolling down an inclined plane without slipping: (a) A hollow cylinder of mass m and inner and outer radii TI and r2j respectively. (b) A hoop of mass m and radius r. (c) A sphere of mass m and radius r. (d) A hollow sphere of mass m and inner and outer radii r^ and r2, respectively. (e) A spherical shell of negligible thickness of mass m and of radius r. (f) Compare the times of descent in (c) and (e). 2. A sphere is projected, without initial rotation, up a perfectly rough inclined plane. Discuss the motion. 3. A wheel which is rotating about its own axis is placed on a perfectly rough inclined plane. Discuss the motion up the plane. "